Teichmüller Spaces and Teichmüller Modular Groups I
The Teichmüller modular groups, also known as the
mapping class groups of surfaces, are the groups of the
isotopy class of self-diffeomorphisms of a surface. At the
same time they may be considered as the orbifold fundamental
groups of the moduli spaces of Riemann surfaces. The moduli
spaces are the quotients of Teichmüller spaces by the natural
action of Teichmüller modular groups.
We will describe how these actions can be used in order to understand the algebraic structure of Teichmüller modular groups (for example, to compute their cohomological dimension). A key ingredient is the complex of curves, which describes the combinatorics of degenerations of Riemann surfaces.
For the most of the time, we will approach Teichmüller spaces from the elementary viewpoint of the hyperbolic surfaces (as opposed to the complex-analytic point of view).
We will also make a connection with a topic discussed in the lectures of S. Wolpert, namely with the classical result of Royden which inspired the Masur-Wolf theorem about the WP isometries.